The result of these computations should be a matrix with the value of $\detname{A}$ in the diagonal entries and zeros elsewhere. The suggestion of using a nonsingular matrix was partially so that it was obvious that the value of the determinant appears on the diagonal.
This result (which is true in general) provides a method for computing the inverse of a nonsingular matrix. Since $A\transpose{C}=\detname{A}I_n$, we can multiply by the reciprocal of the determinant (which is nonzero!) and the inverse of $A$ (it exists!) to arrive at an expression for the matrix inverse:
\begin{equation*}
\inverse{A}=\frac{1}{\detname{A}}\transpose{C}
\end{equation*}